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Modular 3

Modular 3. Ch 2.2 to 2.4. Ch 2.2 Organizing Quantitative Data : The Popular Displays . Objective A : Histogram. Objective B : Using StatCrunch to Construct a Histogram . Objective C : Constructing a Stem-and-Leaf Plot.

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Modular 3

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  1. Modular 3 Ch 2.2 to 2.4

  2. Ch 2.2 Organizing Quantitative Data : The Popular Displays Objective A : Histogram Objective B : Using StatCrunch to Construct a Histogram Objective C : Constructing a Stem-and-Leaf Plot Objective D : Construct Frequency Distributions and Histogram for Continuous Data Ch2.3 More Graphical Displays & Misleading Graphs Objective A : Polygon Objective B : Cumulative Frequency Table and Ogive Objective C : Time Series Graphs Ch 2.4 Graphical Misrepresentations of Data

  3. Ch 2.2 Organizing Quantitative Data : The Popular Displays Objective A : Histogram A histogram is constructed by drawing rectangles for each class of data. If the discrete data set is small, each number is a class. If the discrete data set is large or the data are continuous, the classes must be created using interval of numbers. The height of each rectangle is the frequency or relative frequency of the class. The width of each rectangle is the same and the rectangles touch each other.

  4. Construct Frequency Distribution and Histogram for Discrete Data Example 1 : The following data represent the number of customers waiting for a table at 6:00 p.m. for 40 consecutive Saturdays at Bobak’s Restaurant: 11 5 11 3 6 8 6 7 4 5 13 9 6 4 14 11 13 10 9 6 8 10 9 5 10 8 7 3 8 8 7 8 7 9 10 4 8 6 11 8 (a) Are these data discrete or continuous? Explain Discrete because the data are whole numbers.

  5. (b) Construct a frequency distribution of the data.

  6. (c) Construct a relative frequency distribution of the data.

  7. (d) What percentage of the Saturdays had 10 or more customers waiting for a table at 6:00 p.m.? 0.100 + 0.100 + 0.000 + 0.050 + 0.025 = 0.275  0.3 (e) Construct a frequency histogram of the data.

  8. Identify the shape of a distribution. Bell-shaped curve Uniform distribution Right-skewed distribution Left-skewed distribution

  9. Ch 2.2 Organizing Quantitative Data : The Popular Displays Objective A : Histogram Objective B : Using StatCrunch to Construct a Histogram Objective C : Constructing a Stem-and-Leaf Plot Objective D : Construct Frequency Distributions and Histogram for Continuous Data Ch2.3 More Graphical Displays & Misleading Graphs Objective A : Polygon Objective B : Cumulative Frequency Table and Ogive Objective C : Time Series Graphs Ch 2.4 Graphical Misrepresentations of Data

  10. Objective B : Using StatCrunch to Construct a Histogram

  11. Ch 2.2 Organizing Quantitative Data : The Popular Displays Objective A : Histogram Objective B : Using StatCrunch to Construct a Histogram Objective C : Constructing a Stem-and-Leaf Plot Objective D : Construct Frequency Distributions and Histogram for Continuous Data Ch2.3 More Graphical Displays & Misleading Graphs Objective A : Polygon Objective B : Cumulative Frequency Table and Ogive Objective C : Time Series Graphs Ch 2.4 Graphical Misrepresentations of Data

  12. Objective C : Constructing a Stem-and-Leaf Plot The stem of a data value will consist of the digits to the left of the rightmost digit. The leaf of a data value will be the rightmost digit.

  13. Example 1 : The following data represent the number of miles per gallon achieved on the highway for small cars for the model year 2008. 27 31 28 30 52 25 33 33 29 23 27 37 30 45 24 32 34 35 31 44 42 26 43 35 36 36 54 33 32 35 34 37 (a) Construct a stem-and-leaf plot. Stem Leaf 2 3 4 5 6 7 7 8 9 3 0 0 1 1 2 2 3 3 3 4 4 5 5 5 6 6 7 7 4 2 3 4 5 5 2 4 (b) Describe the shape of the distribution. Slightly skewed to the right.

  14. Ch 2.2 Organizing Quantitative Data : The Popular Displays Objective A : Histogram Objective B : Using StatCrunch to Construct a Histogram Objective C : Constructing a Stem-and-Leaf Plot Objective D : Construct Frequency Distributions and Histogram for Continuous Data Ch2.3 More Graphical Displays & Misleading Graphs Objective A : Polygon Objective B : Cumulative Frequency Table and Ogive Objective C : Time Series Graphs Ch 2.4 Graphical Misrepresentations of Data

  15. Objective D : Construct Frequency Distributions and Histogram for Continuous Data • Classes are the categories by which data are grouped. • The lowest class limit is the smallest value within the class. • The upper class limit is the largest value within the class. • The class width is the difference between consecutive lower class limits. • The class width is computed by the following formula. largest data value – smallest data value Class width number of class -------> Round this value up

  16. Example 1 : The following data represent the fall 2006 student headcount enrollments for all public community colleges in the state of Illinois. (a) Find the number of class. 6

  17. (b) Find the class limits. Lowest class limits: 0, 5,000, 10,000, 15,000, 20,000, 25,000 Upper class limits: 4,999, 9,999, 14,999, 19,999, 24,999, 29,999 (c) Find the class width. Class width = 5000 – 0 = 5000

  18. Example 2 : The following data represent the percentage of people without health insurance for the 50 states and the District of Columbia in 2006. With the first class having a lower class limit of 24000 and a class width of 3000 :

  19. (a) Construct a frequency distribution. Put the data in ascending order first : ( ) ( 24360 25112 25204 25792 26104 26406 26754 26839 27419 27763 28019 28185 28339 28553 28777 28895 28979 29066 29223 29310 29456 29515 29808 30117 30317 30439 30676 30935 31012 31116 31635 31856 32222 32290 32734 33334 33373 33419 33494 33595 33628 33683 34332 34964 35407 36176 37574 38794 39840 40973 47515 ) ( ( ) ) ( ) ( ) ( )

  20. (b) Construct a relative frequency distribution.

  21. (c) Construct a frequency histogram of the data.

  22. (d) Construct a relative frequency histogram of the data. (e) Describe the shape of the distribution. Skewed to the right.

  23. (f) Repeat parts (a) and (c) using a class width of 4000. Frequency distribution: ( 24360 25112 25204 25792 26104 26406 26754 26839 27419 27763 28019 28185 28339 28553 28777 28895 28979 29066 29223 29310 29456 29515 29808 30117 30317 30439 30676 30935 31012 31116 31635 31856 32222 32290 32734 33334 33373 33419 33494 33595 33628 33683 34332 34964 35407 36176 37574 38794 39840 40973 47515 ) ( ) ( ) ( ) ( ) ( )

  24. Frequency histogram :

  25. (g) Does one frequency distribution provide a better summary of the data than the other? Explain. The frequency distribution of the smaller class width provides a better details of the data than the frequency of bigger class width. But the shapes of the both curves are similar.

  26. Example 3 : The largest value of a data set is 125 and the smallest value of the data set is 27. If six classes are to be formed, calculate a class width and list the lower and upper class limit for each class. largest data value – smallest data value = Roundup ( ) Class width number of class 125 – 27 = Roundup ( ) 6 = Roundup( 16.3333) 17

  27. Ch 2.2 Organizing Quantitative Data : The Popular Displays Objective A : Histogram Objective B : Using StatCrunch to Construct a Histogram Objective C : Constructing a Stem-and-Leaf Plot Objective D : Construct Frequency Distributions and Histogram for Continuous Data Ch2.3 More Graphical Displays & Misleading Graphs Objective A : Polygon Objective B : Cumulative Frequency Table and Ogive Objective C : Time Series Graphs Ch 2.4 Graphical Misrepresentations of Data

  28. Ch2.3 More Graphical Displays Objective A : Polygon A1. Class Midpoints • A class midpoint is the sum of consecutive lower class limits divided by 2. Example 1 : The following frequency distribution represents a random sample of 40 car speeds for crossing an intersection in Sylmar. Find the class midpoints of each class.

  29. Find the class midpoints of each class.

  30. A2. Frequency of Polygons A frequency polygon is a graph that uses line segments connect to points directly above the class midpoint value. The heights of the points are the class frequencies. Two additional line segments are drawn connecting each end of the graph with the horizontal axis.

  31. Example 2 : Construct a frequency polygon for example 1. From the table of example 1:

  32. Example 3 (p.105 #6) : Deaths by Legal Intervention Deaths by legal intervention refers to injuries inflicted by law-enforcement agents in the course of arresting or attempting to arrest lawbreakers, suppressing disturbances, maintaining order, and other legal action (including legal execution). In 2006, 174 such deaths occurred in 16 states in the United States. The following frequency polygon represents these deaths by age.

  33. (a) What is the class width? How many classes are represented in the graph ? Class width is 10. 9 classes are represented in the graph. (b) What is the midpoint of the first class? What are the lower and upper limits of the first class? The midpoint of the first class is 5. The lower and upper limits of the first class are 0 – 9. (c) What is the midpoint of the last class? What are the lower and upper limits of the last class? The midpoint of the first class is 85. The lower and upper limits of the first class are 80 – 89.

  34. (d) Which age group had 35 deaths due to legal intervention? 40 – 49 (e) Which two age groups have the highest number of deaths due to legal intervention? Estimate the number of deaths for these age groups. 20 – 29 and 30 – 39 have the highest number of deaths due to legal intervention. 46 number of deaths for these age groups. (f) Estimate the relative frequency for the class 20 – 29.

  35. Ch 2.2 Organizing Quantitative Data : The Popular Displays Objective A : Histogram Objective B : Using StatCrunch to Construct a Histogram Objective C : Constructing a Stem-and-Leaf Plot Objective D : Construct Frequency Distributions and Histogram for Continuous Data Ch2.3 More Graphical Displays & Misleading Graphs Objective A : Polygon Objective B : Cumulative Frequency Table and Ogive Objective C : Time Series Graphs Ch 2.4 Graphical Misrepresentations of Data

  36. Objective B : Cumulative Frequency Table and Ogive B1. Cumulative Frequency/Cumulative Relative Frequency Table • For continuous data, a cumulative frequency table displays the total number of observations less than or equal to the upper class limit of a class. • For continuous data, a cumulative frequency table displays the percentage of observations less than or equal to the upper class limit of a class.

  37. Example 1 : Construct a cumulative frequency table for the data summarized below.

  38. B2. Ogive An ogive is a graph that represents the cumulative frequency or cumulative relative frequency for the class. It is constructed by plotting points whose x-coordinates are the upper class limits and whose y-coordinates are the cumulative frequencies or cumulative relative frequencies of the class. Then line segments are drawn connecting consecutive points. And additional line segment is drawn connecting the first point to the horizontal axis at a location representing the upper limit of the class that would precede the first class (if it existed).

  39. Example 1 : Construct a ogive for the previous example. Cumulative Frequency Table : Cumulative Frequency Ogive:

  40. Ch 2.2 Organizing Quantitative Data : The Popular Displays Objective A : Histogram Objective B : Using StatCrunch to Construct a Histogram Objective C : Constructing a Stem-and-Leaf Plot Objective D : Construct Frequency Distributions and Histogram for Continuous Data Ch2.3 More Graphical Displays & Misleading Graphs Objective A : Polygon Objective B : Cumulative Frequency Table and Ogive Objective C : Time Series Graphs Ch 2.4 Graphical Misrepresentations of Data

  41. Objective C : Time Series Graphs A time series graph represents the values of a variable that have been collected over a specified period of time. The horizontal axis is the time and the vertical axis is the value of the variable. Line segments are drawn by connective consecutive points of time and corresponding value of the variable.

  42. Example 1: The following time-series graph shows the annual U.S. motor vehicle production from 1990 through 2008.

  43. (a) Estimate the number of motor vehicles produced in the United States in 1991. 8900 thousands vehicles. (b) Estimate the number of motor vehicles produced in the United States in 1999. 13000 thousands vehicles. (c) Use the results from (a) and (b) to estimate the percent increase in the number of motor vehicles produced from 1991 to 1999. (d) Estimate the percent decrease in the number of motor vehicles produced from 1999 to 2008.

  44. Ch 2.2 Organizing Quantitative Data : The Popular Displays Objective A : Histogram Objective B : Using StatCrunch to Construct a Histogram Objective C : Constructing a Stem-and-Leaf Plot Objective D : Construct Frequency Distributions and Histogram for Continuous Data Ch2.3 More Graphical Displays & Misleading Graphs Objective A : Polygon & Ogive Objective B : Time Series Graphs Ch 2.4 Graphical Misrepresentations of Data

  45. Ch 2.4 Graphical Misrepresentations of Data The most common graphical misinterpretation of data is accomplished through manipulation of the scale of the graph. Example 1 : Union Membership The following relative frequency histogram represents the proportion of employed people aged 25 to 64 years old who were members of a union.

  46. (a) Describe how this graph is misleading. What might a reader conclude from the graph? The vertical axis starts a 0.08 instead of 0. Readers may think the proportion of those employed aged 45 to 54 years who are union members is much higher than for those aged 35 to 44 years. (b) Redraw the histogram so that it is not misleading

  47. Example 2 : Inauguration Cost The following is a USA Today-type graph. Explain how it is misleading. The lengths of the bars are no proportional. For example, the bar representing the cost of Clinton’s inauguration should be slightly more than 9 times the one for Carter’s cost and twice as long as the bar representing Reagan’s cost.

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